Simple coloring, which you taught me, is something I understand, although I often don't visualize things so I can execute it. However, in this example, the same exclusion is also accomplished with strong links, or a "fork."

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Lets pretend the cells in row 3 are not strongly linked. We would have two seperate chains (A,a) and (B,b), which are connected by a weak link in row 3. We know that A & B both can't be 5, so at leat one of a & b must be. Any cell that 'sees' a & b can't be 5. That's multi-colouring.

Here I don't understand the "let's pretend" bit. If row 3 wasn't strongly linked, then how could we have the A,a chain? Be that as it may, this looks the same as the simple coloring above, except we have a B,b substituted for an A,a.

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From the same position in the puzzle, there is multi-colouring on 1 that excludes 1 from r1c2 and r6c3. Two chains weakly linked in column 4 with b & a, so at least one of A & B must be 1.

The above are two examples of multi-coloring which I am able to validate by doing tests via little chains. However, I've seen many examples, but have never learned any rules or principles which I could use to actually use this technique.

Let's see if I can derive some from the examples.

1) We make two chains
2) The chains must be weakly linked. I assume the weak link must mean a third occurrence of the number in the row/column/box.
3) Then the cells that must contain the number are of the opposite polarity of those in the weak link. That is, if the chain segment in the weak link is a "+", then the actual occurrence of the number must be in a "-" cell and vice versa.

Suppose the cells marked * are the only occurrences of a candidate in R1, R7, and C6. There are three strong links you can connect. If you color the cells in the chain alternately blue and red, then either the blue cells are all true, or the red ones are. One of the cells marked * in B1 and B7 will be blue, the other red. ONE of them must be true, and you can eliminate the candidate value in all the cells marked #. This is coloring.

The strong link in C6 is gone. But, we can make the same eliminations! Depending on your book of worship, this is a fork, a skyscraper, or a Turbot fish. More generally, this is perhaps the simplest example of multicoloring.

ONE or BOTH of the cells marked * in B1 and B7 will be true.

Let's pick one of the # cells as one where an elimination occurs, and relabel the diagram:

If A is true, c is not true.
If A is not true, a is true, b is false, B is true, and c is not true. Either way, c is not true. Note that if d is true, both A and B are true.

You can draw a diagram: c-A-a-b-B-c. This 5-sided creature is a Turbot fish.

You can also contemplate some kind of "rule" which allows you to connect the strong links (A-a and B-b) across a weak link, a-b. I don't have such a rule. I argue the logic above each time I encounter these patterns.

My rule of thumb is: Make diagrams of connected strong links. If the resulting chains have an odd number (1,3, ...) of links, you may find an elimination by connecting them across a weak link.

2) The chains must be weakly linked. I assume the weak link must mean a third occurrence of the number in the row/column/box.

Yes

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3) Then the cells that must contain the number are of the opposite polarity of those in the weak link. That is, if the chain segment in the weak link is a "+", then the actual occurrence of the number must be in a "-" cell and vice versa.

No.

You can say nothing about where the actual occurrence of the number is. You can only say that both of the weakly linked cells cannot be true. Therefore, at least one of the cells of the opposite polarity in each chain must be true. Any cell that 'sees' both opposite polarity cells can have the number excluded.

When two seperate chains are weakly linked, they function as a single chain. With simple colouring (my first example) we have one chain. We know that either A is true or a is true, so any cells thats 'sees' both ends of the chain cannot be true. With multi-colouring (my made-up second example), we have two chains that are weakly linked by one of the ends of each chain. We know that the weakly linked ends BOTH can't be true, so either the opposite end of one chain is true or the opposite end of the other chain is true. Now the situation is just like simple colouring: Any cell that 'sees' both those ends can't be true.

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However, in this example, the same exclusion is also accomplished with strong links, or a "fork."

Not sure where the term "fork" comes from, but strong links is nothing more that colouring, so it's no surprise that they make the same exclusion.

Suppose the cells marked * are the only occurrences of a candidate in R1, R7, and C6. There are three strong links you can connect. If you color the cells in the chain alternately blue and red, then either the blue cells are all true, or the red ones are. One of the cells marked * in B1 and B7 will be blue, the other red. ONE of them must be true, and you can eliminate the candidate value in all the cells marked #. This is coloring.

The strong link in C6 is gone. But, we can make the same eliminations! Depending on your book of worship, this is a fork, a skyscraper, or a Turbot fish. More generally, this is perhaps the simplest example of multicoloring.

If A is true, c is not true.
If A is not true, a is true, b is false, B is true, and c is not true. Either way, c is not true. Note that if d is true, both A and B are true.

You can draw a diagram: c-A-a-b-B-c. This 5-sided creature is a Turbot fish.

You can also contemplate some kind of "rule" which allows you to connect the strong links (A-a and B-b) across a weak link, a-b. I don't have such a rule. I argue the logic above each time I encounter these patterns.

Keith, I'm getting more confused, not with the logic, but with the terminology. In the first grid, we have strong links (which I thought was also called fork or skyscraper) AND a simple coloring chain, either of which will make the eliminations.

In the next two grids, the third occurrence in c6 seems to remove the coloring chain possibility, but I see it still as a case of simple strong links. However, we're talking about multi-coloring and turbot fish while I see nothing more than the aforementioned strong links to eliminate those candidates in b17. What am I missing?

Strong links = only two cells in a group (row, column, box) that can possibly contain a single digit = conjugate links.

Strongly linked chain = simple colouring chain.

Simple colouring chain = alternately labeled strongly linked cells (chain must be composed of at least 4 cells in order to make any exclusions, but can be of any length).

turbot fish = a seni-specific arrangement, or pattern, of cells that compose a strongly linked chain.

skyscraper = a semi-specific arrangement, or pattern, of cells that compose a strongly linked chain.

fork = a semi-specific arrangement, or pattern, of cells that compose a strongly linked chain.

turbot fish = fork = skyscraper.

turbot fish + skyscraper + fork = subset of strongly linked chain/simple colouring chain. (Every turbot fish + skyscraper + fork can be described as colouring, but not every instance of colouring is a turbot fish/skyscraper/fork.)

weak link = more that two cells in a group (row, column, box) that can possible contain a single digit

multi-colouring = two simple colouring chains connected by a weak link.
(Too add to the confusion, turbot fish/skyscraper/fork can also be considered as a subset of multi-colouring, because their exclusions work whether they are a single chain or two chains connected by a weak link.)

Marty R. wrote:

In the next two grids, the third occurrence in c6 seems to remove the coloring chain possibility, but I see it still as a case of simple strong links.

The diagram below shows a single chain of strongly linked cells on the digit 5, alternately label (A) and (a). This is also know as simple colouring because only one chain is involved. Any cell that 'sees' both (A) and (a) can have the 5 removed, which are the cells marked #. This particular pattern can also be called fork, skyscraper & turbot fish.

Two seperate, strongly linked chains but connected by the weak link in column 6. This is now called multi-colouring, because two chains are involved. The exclusions are the same as for a simple colouring chain.
This particular pattern can also be called fork, skyscraper & turbot fish.

The original description/method of nishio is the following: For a grid focused on a single digit. Assume that digit occupies a certain start cell. If, from that assumption, you cannot then place that digit in every group, then you can remove that digit as an option from that start cell. From this description, it is easy to see why nishio is considered a brute force or T&E method.

With all due respect, MJ, I don't think this is easy to see, at all. Consider the following example.

Now, the thought process that says "If I place a "7" at r3c8, I will not be able to complete the puzzle, so the "7" in column 8 must appear in row 2" fits the definition of nishio exactly as you gave it. Does that thought process constitute "trial and error"?

Well of course it is T&E. So is assuming either r2c8 or r3c2 <> 7 and finding out that the puzzle crashes in those cases. How else could you possibly perform T&E from that position? Are you going to try to argue that you can't perform T&E on that grid?

David Bryant wrote:

I submit that it does not constitute T&E. It's entirely logical, it works just as well as saying "The possible '7' at r2c8 is unique in row 2" and, depending on the image of the grid that exists in the mind of the solver, it may be easier to see.

Well of course it works just as well. Brute force T&E can easily replace any other method, and is usually easier to perform. If that is your criteria for an acceptable method, then go ahead and plug in numbers anywhere and anytime. It does not matter to me in the least.

David Bryant wrote:

Myth Jellies wrote:

The logic behind hinges can be used by nishio, but they can also be accomplished via simple grouped coloring or multi-coloring, and thus can be used without ever assuming any digit occupies a particular cell ...

It's not that the "logic behind hinges" can be used by nishio -- it's the exact same logic in either case. The solver who employs nishio says "If I place this digit here, I can't complete the puzzle." The solver who recognizes the "hinge pattern" in effect says "I know that somebody used a proof by contradiction to show that I can't place this digit here." The "hinge" is just a black box that conceals the reductio ad absurdum from the innocent.

Not true at all. Consider the hinge...

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a7 a7 a7
A7 . .
A7 . .

Why would I need a proof by contradiction? It is easy to see that the two groups, 'a' and 'A', span the entire gamut of possible locations for the digit 7 in this box and they do not intersect. This makes them conjugate groups, and so one can freely assign a color to each group or slap a strong or weak link between them without any assumption that either group actually does or does not contain the 7.

Therefore, hinges are not nishio. They are simply useful groupings of digits which may be used by brute force T&E and linked pattern methods alike.

David Bryant wrote:

Moreover, the distinction you're attempting to draw lacks substance. Formal logic tells us that the statements "A" and "not(not A)" always have the same truth value. A proof by contradiction is just as good as any other.

If all you are trying to do is prove your puzzle is a sudoku, then brute force T&E is as good a method as any other. Assume anywhere, anytime. But don't bother trying to tell me that your assumptions are somehow special and the solver who blindly plugs in digits and works out a contradiction or a solution is inferior. Since every basic method, as well as many coloring and chaining and ALS methods can be performed without making any assumption about any particular candidate being true or false; I continue to maintain that performing any step which requires such an assumption is performing brute force T&E.

David Bryant wrote:

Myth Jellies wrote:

Since no assumptions are made, some believe that these templating nishio methods are not as brute force/T&E as the original nishio is.

Imho, anybody who believes that needs a refresher course in Boolean Algebra. All that a "templating method" accomplishes is to hide the details of the proof by contradiction inside a black box.

Oh, foul! Come on, you can't make all these polarizing comments and then say, "That's not what I want to discuss. That's been done to death already." Well, obviously you can, but....

David Bryant wrote:

What I am interested in is understanding how to solve these puzzles quickly and easily....It appears to me that the "nishio" elimination is, in this instance at least, operationally more useful to a human solver. Why let some high-flown rhetoric about "brute force" stand between the solver and an elegant solution?

High-flown rhetoric?Quickest and easiest method is basic techniques to roughly x-wings, uniqueness deadly patterns, a quick scan for xy-wings, and then do brute force T&E assume a number and see if it crashes or solves the puzzle. So if quickness and utility are your overriding concerns, then there you go. Some might get bored with it pretty quickly, as it doesn't require much thought, but it solves everything, and I guess elegance is in the eye of the beholder.

Tracy, thank you for the definitions and other information. I'm going to have to get a better understanding of all these definitions, because despite all the terms, I still see one technique, which I learned as "strong links", that eliminates the candidates.

I am fascinated by David Bryant's and Myth Jellies' exchange, and I think I can sum up the critical distinction.

Myth Jellies wrote:

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Therefore, hinges are not nishio. They are simply useful groupings of digits which may be used by brute force T&E and linked pattern methods alike.

How about, rather, all hinges contain nishio, but not all nishio contain hinges.

It seems to me that we are all talking about patterns of strong links. When David points out a "double implication chain," it is a pattern of strong links in bivalue cells, sometimes involving one value, sometimes more than one value.

In the way that Myth Jellies has chosen to define "brute force T&E," the solving method "contains" emminently logical possibilities when the decision to start a chain is based on a pattern of strong links. I believe what Myth Jellies objects to is using chains to reveal the patterns of strong links. MJ wants to know the answer before we ask the question, which is fine.

What David is saying, I think, is that all we need is to know is that there is an answer, not necessarily what it will be, which is also fine.

Logic, like chess, is both an art and a science; so is astrophysics, quantum mechanics, and Tarot reading.

think, there is no doubt, that i appreciate your POV. And there is no doubt that mine is nearer to David Bryant's.

Myth Jellies wrote:

So if quickness and utility are your overriding concerns, then there you go. Some might get bored with it pretty quickly, as it doesn't require much thought, but it solves everything, and I guess elegance is in the eye of the beholder.

So i dont think so. We also have a sense for elegance (maybe not for the same as you), thought and fun

Keith, I'm getting more confused, not with the logic, but with the terminology. In the first grid, we have strong links (which I thought was also called fork or skyscraper) AND a simple coloring chain, either of which will make the eliminations.

In the next two grids, the third occurrence in c6 seems to remove the coloring chain possibility, but I see it still as a case of simple strong links. However, we're talking about multi-coloring and turbot fish while I see nothing more than the aforementioned strong links to eliminate those candidates in b17. What am I missing?

I think my point is that these things are all the same. I, like you, would identify the two strong links. Then, if they line up at one end, look for the implications. Fork, multicoloring, Turbot fish, whatever.

When I first was studying strong links, I noticed that earlier than Havard's "skyscraper", the formation had been called a fork. I have used the earlier term not only because first named wins, but because I think it is more descriptive.

The difference is: You can connect, across weak links, strong chain segments that have an odd number of links, to possibly get a useful result. You cannot usefully connect (across a weak link) strong chain segments if any has an even number of links.

(I'm going to post this now. I may have to edit it later. It's been a long day, and the rest of my week will be no better.)